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One Cube Estimate Implies the Yang--Mills Mass Gap

A mix of human-written and AI-generated textHuman understanding: all partsmath-ph — Mathematical Physicsmath.PR — Probability

Contributed by Deep Bhattacharjee, Ushashi Bhattacharya ↗, James Montgomery Lockwood, Tristan Romanov

Version 7 / Oct 18, 2026 / CC BY 4.0

Abstract

We prove that four-dimensional lattice Yang--Mills theory with any compact simple gauge group has a unique infinite-volume state and a mass gap at strong coupling, and that one averaged estimate in a single cube of fixed physical size gives a mass gap to every continuum limit. For $\SU(2)$ the block variables approach the Haar measure in every representation as the coupling is tuned to the continuum, and this proves the estimate at the coarsest scale on every torus of side up to e2k/(17k)e^{2^k/(17k)} after kk blocking steps. The estimate along the whole tuning remains open.

Author comments: Retitled "One Cube Estimate Implies the Yang--Mills Mass Gap". Adds the labels and islands of Balaban's blocks, the averaged cube condition and its passage across scales, the smoothing along paths for SU(2), the metrics that compose across scales, the conditional chain for every compact simple group and a concluding section; every constant is proved by hand. Source, programs and Lean 4 certificate: https://github.com/creelie/4Y-M (release v4.37.0, doi:10.5281/zenodo.23992883). The paper does not prove the cube estimate along the whole continuum tuning or the existence of a nontrivial continuum limit, so it does not solve the Clay Millennium problem.

Provenance statement

Deep Bhattacharjee conceived the project and, with the help of AI agents working under his direction (Claude Opus 5.5 in Claude Code, Claude Fable 5.1, GPT-6.1 Sol Ultra and GPT-6 Astra Ultra), made its constructions and proved its main results: the forest gauge on the period-two line field and the free-boundary state with its transfer-matrix gap at strong coupling; the plaquette blocks of SU(2) and SU(3) with their constants proved by hand; the reduction of the mass gap to one averaged estimate in a cube of fixed physical size; the labels of Bałaban's blocks and the islands of bad blocks; the averaged condition, its descent to block variables, the upward step and the split inequality; the single-link coefficients across scales, the forces in the fibre law, the smoothing along the paths for SU(2) and the row sums of the Gaussian model; and the conditional chain for every compact simple group. He wrote the programs of the certified computations and the paper, and checked every statement. Ushashi Bhattacharya checked the proofs and the references and edited the text. James Montgomery Lockwood contributed the analysis of constrained cubes and its code and Lean 4 formalisations, and the composition of metrics across scales of Section X S with its Lean 4 formalisation; Tristan Romanov contributed the certified bound on the forest-gauge Hessian, the research continuation on which Section XIII is built and the flat fields far from the identity, with four Lean 4 modules. All four authors approved the final text. The Anaxia engine of the BLLM system first assembled the combination of the forest gauge with the sharp quarter behind the SU(2) theorem (doi:10.5281/zenodo.23469389; run records doi:10.5281/zenodo.23484160) and supplied no step of the proofs. The paper does not prove the cube estimate along the whole continuum tuning or the existence of a nontrivial continuum limit, so it does not solve the Clay Millennium problem, and it says so. Compared with version 6 (hexagonmath.org/2610.00341v6), the paper adds Section XIII (labels of the blocks, islands of bad blocks, the bad side of a projected block average, the obstructions to the remaining routes, the effective action across one scale), Sections X F to X S (an averaged form of the cube condition, its local form, the couplings where it holds and fails, its neutral part on small tori, its passage to block variables, the upward step, the measures for which it holds, the split inequality at strong coupling, the condition at the coarsest scale, the coefficients across scales, the forces in the fibre law, the smoothing along the paths, the order of the row sums of the Gaussian model and the metrics that compose across scales), Sections XI D, XI H and XI I (the Osterwalder–Schrader axioms for the flowed fields, the limit at separated points of the fields without flow with the inputs it needs, and the conditional chain for every compact simple group) and the concluding Section XIV, and it proves by hand every constant that version 6 took from computer certificates; the title-page footnote says so. Tools used: Anthropic Claude Opus 5.5 (Claude Code) and Claude Fable 5.1; OpenAI GPT-6.1 Sol Ultra and GPT-6 Astra Ultra; BLLM Anaxia engine v3.0.5 and v3.1.0; Lean 4 with Mathlib; Python (NumPy, SciPy, mpmath, python-flint with FLINT and Arb); C; shell. arXiv identifier: none. Formalization: https://github.com/creelie/4Y-M/tree/main/YangMills (release v4.37.0, https://doi.org/10.5281/zenodo.23992883 (all versions https://doi.org/10.5281/zenodo.23444338))
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Tools used

Anthropic
Claude CodeVersion Fable 5.1 Medium

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Version history

  1. v1 · 29 pagesSubmitted by Deep BhattacharjeeInitial depositSupersededOct 18, 2026
  2. v2 · 28 pagesSubmitted by Deep BhattacharjeeSubmitted as a new versionSupersededOct 22, 2026
  3. v3 · 54 pagesSubmitted by Deep BhattacharjeeSubmitted as a new versionSupersededOct 28, 2026
  4. v4 · 70 pagesSubmitted by Deep BhattacharjeeSubmitted as a new versionSupersededOct 30, 2026
  5. v5 · 150 pagesSubmitted by Deep BhattacharjeeSubmitted as a new versionSupersededOct 31, 2026
  6. v6 · 105 pagesSubmitted by Deep BhattacharjeeSubmitted as a new versionSupersededNov 02, 2026
  7. v7 · 385 pagesSubmitted by Deep BhattacharjeeRetitled "One Cube Estimate Implies the Yang--Mills Mass Gap". Adds the labels and islands of Balaban's blocks, the averaged cube condition and its passage across scales, the smoothing along paths for SU(2), the metrics that compose across scales, the conditional chain for every compact simple group and a concluding section; every constant is proved by hand. Source, programs and Lean 4 certificate: https://github.com/creelie/4Y-M (release v4.37.0, doi:10.5281/zenodo.23992883). The paper does not prove the cube estimate along the whole continuum tuning or the existence of a nontrivial continuum limit, so it does not solve the Clay Millennium problem.CurrentNov 06, 2026